| At present,the research of multi-agent systems(MASs)with cooperative mech-anisms has achieved a lot of rich results.Compared with MASs under cooperative mechanisms,MASs with cooperative-competitive mechanisms can better reveal the complexity of MASs network and have more applications in practical engineering.Rel-evant research has not formed a systematic theoretical system,which motivates us to s-tudy some problems about coordination control of MASs with cooperative-competitive mechanisms in this paper.The main contents are given as follows:1.We focus on studying the consensus tracking control problem of second-order sampled-data MASs under cooperative-competitive mechanisms.Compared with MASs under cooperative mechanisms,the eigenvalues of Laplacian matrix L corre-sponding to the communication topology of MASs under cooperative-competitive mech-anisms cannot be guaranteed to lie in the right half plane.Further,the condition that the leader-following communication topology has a spanning tree is not a necessary and sufficient condition to guarantee the positive stability of matrix H:=L+B(B is a diago-nal matrix to describe the connection relationship between the leader and the follower).By exploring the connection relationship and the range of connection weight between the leader and the follower,we establish a sufficient criterion for the positive stability of matrix H.Then,some sufficient consensus tracking control conditions are obtained for MASs without time-delays and with time-delays by using matrix analysis method and perturbation theory,respectively.Furthermore,the positive stability and eigenvalues of matrix H are difficult to judge and calculate accurately in large-scale networks.There-fore,according to the relationship between the infinite norm and the spectral radius of system matrix,we establish some sufficient consensus tracking control conditions for MASs.2.We study the consensus tracking problem of first-order continuous-time MASs with inconsistent coupling parameters in control protocol.Due to the existence of in-consistent coupling parameters,the positive stability of matrixθH(θis a diagonal ma-trix to describe the coupling strength)is the key factor to affect the consensus tracking of MASs.Firstly,we analyze the relationship between the positive stability of matrix H andθH.Specifically,their positive stability is equivalent under cooperative mech-anisms,but not equivalent under cooperative-competitive mechanisms.Subsequently,we establish some criteria to solve the positive stability of matrixθH.Finally,in view of saving the actual engineering cost,two selection methods of coupling strength ma-trixθare given to ensure that the leader-follower connection number can be as few as possible,or the leader-follower connection weight can be as small as possible on the basis of achieving consensus tracking for MASs.3.The consensus tracking control problem of second-order continuous-time MASs with time-varying delays is studied.Firstly,four equivalent criteria for MASs with fixed topology are given.Then,for MASs with switching topology,using Lyapunov first method and average dwell time method instead of Lyapunov second method,we estab-lish some sufficient consensus tracking conditions for MASs under different cases of subsystems,in which the parameters do not depend on Lyapunov matrix and the upper bound of time-varying delay is less conservative.4.Under cooperative-competitive mechanisms,the group consensus problem of first/second-order discrete-time MASs is studied.Firstly,for matrix P=I-εL,we obtain a larger range ofεand a looser topological condition to guarantee the existence of _kl_→i_+m_∞P~k,which greatly improves the Perron-Frobenius theorem.Subsequently,we can divide all the agents into m subgroups according to the actual demand,and give the design method of weight so that system can achieve group consensus.Finally,we further generalize the results from first-order MASs to second-order MASs. |
